1. Jacek Wallusch _________________________________ Statistics for International Business Lecture 10: Introduction to Hypothesis Testing

2. Getting Started ____________________________________________________________________________________________ Statistics: 10 elements of statistical test nullifying a hypothesis Hypothesis formulation: null hypothesis and alternative hypothesis Null and the data: null hypothesis = research hypothesis alternative hypothesis = converse of null assume valid setting the null hypothesis we assume that the description of nature presented by null hypothesis is valid

3. Getting Started ____________________________________________________________________________________________ Statistics: 10 elements of statistical test nullifying a hypothesis again Test statistic formulation: test statistic – function of the sample measurement Rejection region formulation: rejected a value of the test statistic for which the null hypothesis is rejected

4. Null Hypothesis ____________________________________________________________________________________________ Statistics: 10 examples Equalities and inequalities, two tails or one tail? Single average: [1] Is the actual average sales volume equal to the planned one? [2] Is the average price higher than the cost? [3] Is the average number of complaints smaller than the number of complaint a year ago? Two averages: [1] Is the average sales volume for medium offer larger than the average sales volume for premium offer? [2] Is the average profit-per-employee in Swindon equal to the one in Slough? [3] Is the average salary of female employee smaller than the average male salary?

5. Errors ____________________________________________________________________________________________ Statistics: 10 type I and II True state of nature and tested hypothesis null hypothesis true null hypothesis not true null acceptedcorrect type II error null rejected type I error correct beliefs and statistical practice: human being is capable of err; don’t worry, there are two types of errors

6. Errors ____________________________________________________________________________________________ Statistics: 10 type I and II Example x% of all foreign students would like to stay in Poland less than x% of all foreign students would like to stay in Poland Type I Error Type I Error: null not rejected if less than x% of all foreign students would like to stay in Poland; Type II Error Type II Error: null rejected if x% of all foreign students would like to stay in Poland calculate the probability of commiting an error type I and an error type II

7. Rejection region ____________________________________________________________________________________________ Statistics: 10 formulation Selecting the rejection region select a particular number subject to the expectations......or take a look at the errors

8. Rejection Region ____________________________________________________________________________________________ Statistics: 10 formulation Rejection region: distribution significance level What is the test statistic distribution? What is the significance level rejection region 10% significance level 1% significance level Example: A one-tailed test

9. A bit of practice ____________________________________________________________________________________________ Statistics: 10 rejection region A common practice: 10%, 5% or 1% confidence level Step 1: read carefully the null hypothesis Step 2: run the test Step 3: compare the p-value with the commonly used confidence level

10. p-value ____________________________________________________________________________________________ Statistics: 10 Empirical probability p-value probability of obtaining the estimated test statistic assuming that the null hypothesis is true Values are assigned to probabilities (e.g. 1% significance level) Knowing [1] test statistic distribution [2] test statistic value calculate the probability

11. A bit of practice ____________________________________________________________________________________________ Statistics: 10 test result presentation Summarise the results in a table: assume a confidence level first! describe the distributions along with the degrees of freedom Null hypothesis Test value Critical value* p-valueConclusions Describe the tested hypothesis Null not rejected or null rejected * Null tested at (?)% significance level using the (?) distribution with (?) degrees of freedom

12. Testing Procedures ____________________________________________________________________________________________ Statistics: 10 examples Single Average: Hypotheses two-tailed one-tailed one-tailed test statistic: Excel: ROZKŁAD.T.ODW( , df) Student-t with T-1 d.f. distribution:

13. Population variance ____________________________________________________________________________________________ Statistics: 10 two-tailed tests Large Sample: hypotheses test statistic: distribution:

14. Two Means ____________________________________________________________________________________________ Statistics: 10 two-tailed tests Large Sample: hypotheses test statistic: standard normal distribution: